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Revision Notes
Electron Devices
Electron Drift Velocity
Vd   n
Where mu-sub-n is the mobility of electrons (defined for the material) and epsilon is the
electric field.
Contact Potential Equation:
kT  N a N d 

ln 
q  ni2 
Where V0 is contact potential
kT/q is 25mV (at 300K)
Na is concentration of acceptors (holes) on p-side
Nd is concentration of donors (electrons) on n-side
ni is the intrinsic carrier concentration.
V0 
Fermi-Dirac Distribution function (first term’s work)
f (E) 
1
 E  Ef
1  exp 
 kT



Diode Equation

 qV

I D  I 0  exp  D  1 
 kT


Diode Capacitance
Cj 

W
Junction capacitance, where W is the width of the depletion/transition region and
epsilon is the permittivity.
This is dominant during reverse bias.
qV
C S  e kT
Storage capacitance, varies with forward bias voltage. Normally dominant in forward
bias, and used in voltage-variable capacitors (varicap diodes).
Page 1 of 16
Communications (Tim Tozer)
LCR parallel circuit:
0 
Q
1
LC
f
R
 0
L BW
Amplitude Modulation
e  A0 (1  m sin(  mt )) sin(  c t )
e = modulated signal
m = modulation index (between 0 and 1, usually 0.3)
Nyquist’s sampling theorem
Sampling frequency = 2 x maximum frequency
Quantisation noise power
(v) 2
Pqn 
12
Delta-v is the step size.
Thermal noise power
N  kTB
N= noise power in watts
k = Boltzmann’s Constant = 1.38 x 10-23 J/K = -229 dB J/K
T = noise temperature in Kelvin
B = bandwidth in Hz
This equation does assume a precisely matched load.
Page 2 of 16
Maths (Ken Todd- term 2)
Total Differential
dz z dx z dy


dt x dt y dt
Where z is a function of x and y and x and y are functions of t.
Change of Variables
If z is a function of x and y, and x and y are both functions of u and v, then:
z z x z y


u x u y u
And, similarly
z z x z y


v x v y v
Taylor’s Series in two variables
f ( x, y )  f (a, b)  Df (a, b) 
D 2 f ( a, b) D 3 f ( a, b)

...
2!
3!


k
x
y
h  xa
Dh
k  y b
Stephenson’s method
Treat h and k as constants, to get the series in terms of h and k
Substitute h and k for a,b,x & y.
Page 3 of 16
Finding Stationary points
At a stationary point:
f
0
x
f
0
y
To check for a saddlepoint, use the Del operator:
2
 2 f  2 f 2 f
  2  2
  
x
y
 xy 
If >0 then you have a saddlepoint. Otherwise, check the partials for a maximum or
minimum.
Double Integrals
A Double Integral is the volume under a surface, as defined by four limits.

x b
xa
dx 
y d
y c
f ( x, y )dy
Page 4 of 16
Static Fields
Electrostatics:
Coulomb’s Law
F
Q1Q2
4r 2
Electric Field due to a point charge
E
Q
4r 2
Electric Field due to a line charge

2r
Rho is charge density in Cm-1
E
Electric Field due to an infinite sheet of charge

2
Rho is charge density in Cm-2
E
P.D. between two points
V  Ed
Where E is electric field strength and d is distance
Capacitance of two sheets
C
A
d
where A is the area of the sheets and d is their separation.
Energy stored in a capacitor:
E  12 CV 2
Electric Flux Density:
D E
Page 5 of 16
Displacement current density:
JD 
dD
dt
Conduction current density:
J
dQ
dt
Generalised form of Ohm’s law:
J  E
where sigma is conductivity.
Magnetic fields:
Biot-Savart Law:
 Il  rˆ
B
4r 2
Faraday’s Law:
V 
d
Bd A
dt 
Inductors
Total flux in an inductor (in Webers)
   Bds  kNI
where k is a geometric constant.
Voltage induced in inductor:
V  N
d
dI
 kN 2
dt
dt
Self-inductance (in Henrys)
L  kN 2
V  L
dI
dt
Page 6 of 16
Transistors
Diode equation (approximate form)
 qV 
I D  I S exp  D 
 kT 
Small-signal diode resistance
25mV
ID
ID is the DC bias current.
rD 
Transistor large-signal behaviour:
IC  0 I B
Transistor small-signal behaviour:
IE
25mV
1
re 
gm
gm 
r  1   re
Common-emitter amplifier:
Voltage gain
Av   g m RC
Input resistance
rin  RB1 // RB 2 // r
Output resistance
rout  RC
Overall gain of CE amplifier with source and load resistance:
RL rin Av
VL

VS (rs  Rin )( ro  RL )
Page 7 of 16
Common-collector amplifier
Voltage gain (just less than unity)
g m RE
Av 
1
1  g m RE
Input resistance (very high)
rin  rT // RB1 // RB 2
rT  (1  g m RE )r
Output resistance (very low)
ro  RE // r // re  re
Bear in mind that, in a CC amplifier, changing the load resistance changes the input
resistance of the amplifier. Changing the source resistance changes the output resistance.
Common-base amplifier:
AV  g m RC
Very low input impedance and moderate output impedance
rin  r // re // RE
rout  Rc
Amplifier characteristic summary (ignoring source and load resistances):
Configuration
Voltage gain
Current gain
CE
CC
Av
-gmRC
1
AI


CB
gm R C
1
Input resistance Output
resistance
rin
rout
RC
r // RBias
(re+RE) //
re
RBias
RC
re
Onset of clipping in single transistor amplifiers
For CE and CB amplifiers:
Clipping on the positive half-cycle occurs when the device switches off completely and
the voltage across RC is reduced to zero.
Clipping on the negative half-cycle occurs when VCE is less than 0.7V and the transistor
saturates.
To get maximum output swing, bias should make VC halfway between VB and VCC.
Page 8 of 16
For CC amplifiers:
The output always clips first on the negative half-cycle. Distortion occurs when the bias
current is equal to the current into RE // RL
The bias current must be large enough to handle the largest desired voltage swing.
Miller’s Effect
This limits the high-frequency behaviour of transistor amplifiers.
Effective capacitance of transistor:

Ceff  C  (1  g m RC )CC

RC  RC // RL
High-frequency cut-off point (-3dB point):
1
fC 

2RS Ceff
Myles’s Transistor Amplifier design guidelines:
Maximum gain from a single-stage amplifier (split-supply): gain MAX  20VCC
VCC
then you need a CC output stage
2I E
25
Source impedance: if RS 
where IE is in mA, then use a CC input stage.
IE
Load impedance: if RL 
Field Effect Transistors
FETs are transconductance devices- the gate voltage controls the source-drain current.
MOSFETs (Metal-oxide-semiconductor field effect transistors) come in two types, each
of which comes in two polarities:
N-channel MOSFETs conduct solely by electrons.
P-channel MOSFETs (less common) conduct solely by holes.
Enhancement-mode MOSFETs have no built-in channel. The channel is induced by the
presence of a positive gate voltage. The threshold voltage VT is positive. With no gate
voltage, the device is switched off (normally-off).
Depletion-mode MOSFETs are the reverse. The channel is built into the silicon and is
controlled by the presence of a negative gate voltage. The threshold voltage is negative.
With no gate voltage, the device is switched on (normally-on). To turn the device off,
apply a negative gate voltage.
Page 9 of 16
MOS Capacitors
Assuming p-type silicon substrate:
Making the metal negative relative to the silicon causes accumulation:


Holes are attracted to region under gate
Bands in silicon bend closer to valence band (gets more p-type)
Making the metal positive relative to the silicon causes depletion


Holes at surface of semiconductor are repelled, causing a depletion region to form.
Bands in silicon bend closer to conduction band (gets less p-type)
Making the metal more postive causes inversion:

Energy bands bend further down

Eventually the bands bend enough to make the surface appear locally n-type.

Strong inversion occurs when the surface is as n-type as the bulk is p-type.
When strong inversion occurs, the device has enough charge carriers to conduct usefully.
This point is termed the threshold voltage (VT)
MOS FET DC behaviour
Drain Current
1

2
I D    VGS  VT VDS  VDS 
2


W
  C
L
Where:
ID is the drain current
 is the device constant (or gain constant), normally specified in questions.
VGS is the gate-source voltage
VT is the threshold voltage (normally defined by manufacturer).
VDS is the drain-source voltage.
W is the channel width
L is the channel length (source-drain distance)
 is the mobility
C is the gate capacitance (varies with oxide thickness)
Drain Current at saturation
1
 (VGS  VT ) 2
2
This is the region used for small-signal amplifiers.
I D ( SAT ) 
Page 10 of 16
MOS FET small-signal AC behaviour
Small signal transconductance
g m   (VGS  VT )
This equation holds when the device is saturated i.e. VDS>VGS-VT
When biasing FETs, remember that the gate current is zero.
Common-source amplifier
Used as a gain stage where very high input impedance is required. Frequently used as
differential input stages in opamps.
Small-signal gain:
AV   g m RD
Input impedance:
Z in  RG
RG may be one resistor (depletion-mode) or two in parallel (enhancement mode)
Output impedance:
Z out  RD
Common-drain amplifier
Used as a voltage buffer. Near unity gain with very high input impedance and very low
output impedance. Usually found at input stages to instrumentation (voltmeters, scopes)
Small-signal gain:
AV  1
Input impedance:
Z in  RG
Output impedance:
Z out 
RS
1  g m RS
Common-gate amplifier
Gain stage with low input impedance- usually used for video or RF systems requiring 50
or 75 ohm input impedance. Also used as current amplifiers.
Small-signal gain:
AV  g m RD
Page 11 of 16
Input impedance:
Z in 
RS
1  g m RS
Output impedance:
Z out  RD
NMOS Logic circuits
NMOS inverter threshold equation
Z pu
Z pd
2

VTd
(Vinv  VTe ) 2
Zpu is the aspect ratio (length-to-width ratio) of the pull-up device.
Zpd is the aspect ratio of the pull-down device.
VTd is the threshold voltage of the depletion mode device (pull-up)
VTe is the threshold voltage of the enhancement mode device (pull-down)
Vinv is the cross-over voltage of the inverter (the point at which it changes state).
The equation gives the ratio of the aspect ratios of the devices.
Transmission Lines
Velocity of wave on line
velocity 
1
LC
Characteristic impedance
Z0 
L
C
Voltage Reflection coefficient
V 
Z L  Z0
Z L  Z0
Voltage Transmission coefficient
 1  V
Standing waves:
If you have sinusoidal excitation and v = 1, then the result is a standing wave:
V  2V0 cos( x)e jt
Page 12 of 16
Reactive behaviour of mismatched transmission line:
Short-circuited line of length l:
 l 

Z in  jZ 0 tan 
 velocity 
1
velocity 
LC

(quarter-wave system)
4

Zero reactance (short circuit) occurs when l  (half-wave system)
2
Infinite reactance occurs when l 
Input impedance of a Tx line with arbitrary termination:
 Z cos( l )  jZ0 sin( l ) 

Z in  Z 0  L
 Z 0 cos( l )  jZ L sin( l ) 

(half wavelength) then Zin = ZL, so if the tx line is an integer number of half2
wavelengths then Zin is independent of Z0.
If l 
If l 
Z2

(quarter wavelength) then Z in  0
4
ZL
Rearranging: Z 0  Z in Z L
This is used for quarter-wave impedance matching: a quarter-wave length of transmission
line of a specified characteristic impedance can act as a matching piece:
If you have a 50 source and a 100 load, then Z 0  50 100  72 . Thus a quarterwave length of transmission line of 72 will match a 50 system to a 100 system, at
the specified frequency.
Page 13 of 16
Transmission Lines & Static Fields (term 3)
Gauss’s Law:
Q
 E  ds  
This boils down to the fact that the total field radiated by a number of point charges
inside an enclosing surface is equal to the sum of all the charges enclosed divided by the
permittivity.
Ampère’s Law

loop
H  dl   I
A number of currents are surrounded by a loop. The total magnetic field on the whole
loop is equal to the total current passing through it.
Field Analysis of Coaxial cables
Capacitance per unit length
2
b
ln  
a
b is the radius of the cable outer.
a is the radius of the cable inner.
 is the permittivity of the dielectric.
C
Inductance per unit length
L
 b
ln  
2  a 
Propagation Velocity
V
1

Characteristic Impedance
Z0 
1
2
 b
ln  
 a
Page 14 of 16
Intrinsic impedance
This is the ratio of electric to magnetic fields in the cable:


Zi 
Field Analysis of Parallel Plate Transmission Line
Capacitance per unit length
C
w
d
w is the width of the plates
d is the plate separation
 is the permittivity of the dielectric.
Inductance per unit length
L
d
w
Propagation Velocity
V
1

Characteristic Impedance
Z0 
d 
w 
Page 15 of 16
Circuits
LC Resonant circuits
Resonant frequency
0 
1
LC
Q-factor (Series LC)
Q
0L
R

1
 0 CR
Q-factor (Parallel LC)
Q
R
  0 CR
0 L
Real, Reactive and Apparent Power
Apparent Power
PApparent  Vrms I rms
Apparent Power is measured in Volt-Amps (VA)
Real Power
PRe al  Vrms I rms cos 
Reactive Power
PRe active  Vrms I rms sin 
Page 16 of 16